ESTIMATION OF THE MEAN AND PROPORTION CHAPTER 8 (PART B) ESTIMATION OF THE MEAN AND PROPORTION Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN:  NOT KNOWN 8.3 Three Possible Cases Case I. If the following three conditions are fulfilled: 1. The population standard deviation σ is not known 2. The sample size is small (i.e., n < 30) 3. The population from which the sample is selected is normally distributed. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN:  NOT KNOWN Three Possible Cases Case II. If the following two conditions are fulfilled: 1. The population standard deviation σ is not known 2. The sample size is large (i.e., n ≥ 30) Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN:  NOT KNOWN Three Possible Cases Case III. If the following three conditions are fulfilled: 1. The population standard deviation σ is not known 2. The sample size is small (i.e., n < 30) 3. The population from which the sample is selected is not normally distributed (or its distribution is unknown). Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

ESTIMATION OF A POPULATION MEAN:  NOT KNOWN Three Possible Cases Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

The t Distribution The t distribution is a specific type of bell-shaped distribution with a lower height and a wider spread than the standard normal distribution. As the sample size becomes larger, the t distribution approaches the standard normal distribution. The t distribution has only one parameter, called the degrees of freedom (df). The mean of the t distribution is equal to 0 and its standard deviation is . Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-4 Find the value of t for 16 degrees of freedom and .05 area in the right tail of a t distribution curve. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Table 8.2 Determining t for 16 df and .05 Area in the Right Tail Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Figure 8.6 The value of t for 16 df and .05 area in the right tail. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Figure 8.7 The value of t for 16 df and .05 area in the left tail. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Confidence Interval for μ Using the t Distribution The (1 – α)100% confidence interval for μ is The value of t is obtained from the t distribution table for n – 1 degrees of freedom and the given confidence level. Here is the margin of error of the estimate; that is, Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-5 Dr. Moore wanted to estimate the mean cholesterol level for all adult men living in Hartford. He took a sample of 25 adult men from Hartford and found that the mean cholesterol level for this sample is 186 mg/dL with a standard deviation of 12 mg/dL. Assume that the cholesterol levels for all adult men in Hartford are (approximately) normally distributed. Construct a 95% confidence interval for the population mean μ. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-5: Solution σ is not known, n < 30, and the population is normally distributed (Case I) Use the t distribution to make a confidence interval for μ . Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-5: Solution df = n – 1 = 25 – 1 = 24 Area in each tail = .5 – (.95/2) = .5 - .4750 = .025 The value of t in the right tail is 2.064 Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Figure 8.8 The value of t. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-5: Solution Thus, we can state with 95% confidence that the mean cholesterol level for all adult men living in Harford lies between 181.05 and 190.95 mg/dL. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-6 Sixty-four randomly selected adults who buy books for general reading were asked how much they usually spend on books per year. The sample produced a mean of $1450 and a standard deviation of $300 for such annual expenses. Determine a 99% confidence interval for the corresponding population mean. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-6: Solution σ is not known, n > 30 (Case II) Use the t distribution to make a confidence interval for μ Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-6: Solution df = n – 1 = 64 – 1 = 63 Area in each tail = .5 – (.99/2) = .5 - .4950 = .005 The value of t in the right tail is 2.656 Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Example 8-6: Solution Thus, we can state with 99% confidence that based on this sample the mean annual expenditure on books by all adults who buy books for general reading is between $1350.40 and $1549.60. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

Confidence Interval for μ Using the t Distribution What If the Sample Size Is Too Large? 1. Use the t value from the last row (the row of ∞) in Table V. 2. Use the normal distribution as an approximation to the t distribution. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved